# Properties

 Label 2-20e2-100.39-c0-0-0 Degree $2$ Conductor $400$ Sign $0.876 + 0.481i$ Analytic cond. $0.199626$ Root an. cond. $0.446795$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.309 − 0.951i)5-s + (0.809 + 0.587i)9-s + (1.11 − 1.53i)13-s + (−1.11 + 0.363i)17-s + (−0.809 + 0.587i)25-s + (−0.5 + 1.53i)29-s + (−0.690 + 0.951i)37-s + (0.5 + 0.363i)41-s + (0.309 − 0.951i)45-s − 49-s + (−1.80 − 0.587i)53-s + (0.5 − 0.363i)61-s + (−1.80 − 0.587i)65-s + (1.11 + 1.53i)73-s + (0.309 + 0.951i)81-s + ⋯
 L(s)  = 1 + (−0.309 − 0.951i)5-s + (0.809 + 0.587i)9-s + (1.11 − 1.53i)13-s + (−1.11 + 0.363i)17-s + (−0.809 + 0.587i)25-s + (−0.5 + 1.53i)29-s + (−0.690 + 0.951i)37-s + (0.5 + 0.363i)41-s + (0.309 − 0.951i)45-s − 49-s + (−1.80 − 0.587i)53-s + (0.5 − 0.363i)61-s + (−1.80 − 0.587i)65-s + (1.11 + 1.53i)73-s + (0.309 + 0.951i)81-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$400$$    =    $$2^{4} \cdot 5^{2}$$ Sign: $0.876 + 0.481i$ Analytic conductor: $$0.199626$$ Root analytic conductor: $$0.446795$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{400} (239, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 400,\ (\ :0),\ 0.876 + 0.481i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8349811355$$ $$L(\frac12)$$ $$\approx$$ $$0.8349811355$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + (0.309 + 0.951i)T$$
good3 $$1 + (-0.809 - 0.587i)T^{2}$$
7 $$1 + T^{2}$$
11 $$1 + (-0.309 + 0.951i)T^{2}$$
13 $$1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2}$$
17 $$1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2}$$
19 $$1 + (0.809 - 0.587i)T^{2}$$
23 $$1 + (0.309 - 0.951i)T^{2}$$
29 $$1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2}$$
31 $$1 + (0.809 - 0.587i)T^{2}$$
37 $$1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2}$$
41 $$1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2}$$
43 $$1 + T^{2}$$
47 $$1 + (-0.809 - 0.587i)T^{2}$$
53 $$1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2}$$
59 $$1 + (-0.309 - 0.951i)T^{2}$$
61 $$1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2}$$
67 $$1 + (-0.809 + 0.587i)T^{2}$$
71 $$1 + (0.809 + 0.587i)T^{2}$$
73 $$1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2}$$
79 $$1 + (0.809 + 0.587i)T^{2}$$
83 $$1 + (-0.809 + 0.587i)T^{2}$$
89 $$1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2}$$
97 $$1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$